Multiple scattering in the atmosphere with a rough surface

In the present paper, the intensity of radiation emergent from the atmosphere bounded by a rough surface is discussed with the aid of the superposition method derived by Mukai (1973). The merit of this method is to express the laws of diffuse reflection and transmission for the planetary problem with a rough surface in terms of a scattering and a transmission function for the standard problem.Here the bottom surface is assumed to reflect light in accordance with the slope distribution given by Cox and Munk (1954a, b). The results are discussed in terms of the optical properties and roughness of the bottom surface.     

Reflection and transmission by a vertically inhomogeneous planetary atmosphere

By appealing to the reciprocity principle simple expressions are derived for the plane albedo and the transmissivity of a vertically inhomogeneous, plane parallel atmosphere. The plane albedo is shown to equal the angular distribution of the reflected intensity for isotropie Illumination of unit intensity incident at the top of the atmosphere, while the transmissivity equals the angular distribution of the transmitted intensity for isotropie illumination of unit Intensity incident at the bottom of the atmosphere. Chandrasekhar's solution of the planetary problem (including a Lambert reflecting lower boundary) in terms of the solution to the standard problem (no reflecting ground) is extended to apply to an inhomogeneous atmosphere resting on a surface that reflects radiation anisotropically but with no dependence on the direction of incidence (anisotropic Lambert reflector). The computational aspects are discussed and a procedure for computing the planetary albedo and transmissivity Is outlined for a vertically inhomogeneous, anisotropically scattering atmosphere overlying a partially reflecting surface. Numerical verification and illustration are also provided and it is shown that the assumed vertical variation of the single scattering albedo strongly affects the plane albedo but only weakly the transmissivity.     

Light scattering by an optically thin inhomogeneous spherically symmetric planetary atmosphere

An approximate solution has been obtained for the problem of multiple scattering of light in an optically thin, inhomogeneous spherically symmetric planetary atmosphere illuminated by parallel solar radiation. A three-stream division of the radiation field has been made and a generalized Eddington approximation developed to solve the moment equations of the problem.     

The effect of surface roughness on the transmission of microwave radiation through a planetary surface

To account for surface roughness, the transmission of microwave radiation through a planetary surface to an observer is treated by a Monte Carlo technique. Sizable effects are found near the limb of the planet, and they should be included in analyses of high-resolution observations and high-precision integrated disk observations.     

A method of computing the emergent radiation by the atmosphere in the region ranging from ultraviolet to infrared

A method of computing the diffuse reflection and transmission radiation by an inhomogeneous, plane-parallel planetary atmosphere with internal emission source is discussed by use of the adding method. If the atmosphere is simulated by a number of homogeneous sub-layers, the radiation diffusely reflected or transmitted by the atmosphere can be expressed in terms of the reflection and transmission matrices of the radiation of sub-layers. The diffusely transmitted radiation due to the internal emission source can be also easily computed in the same manner. These equations for the emergent radiation are in a quite general form and are applicable to radiative transfer in the atmosphere in the region from ultraviolet to infrared radiation. With this method, the tiresome treatment due to the polarity effect of radiation is overcome.     

A hybrid mode of diffuse and specular reflector for computation of the emergent radiation by the adding method

A method of computing the diffuse reflection and transmission radiation from an inhomogenous, plane-parallel planetary atmosphere bounded by the hybrid surface of a diffuse and specular reflector is discussed by using the addition method. If the atmosphere is simulated by a number of homogeneous sublayers, the radiation diffusely reflected and transmitted by the atmosphere can be expressed in terms of the diffuse reflection and transmission matrices of radiation of sublayers (Laciset al., 1974; Takashima, 1973, 1975). With this method (Takashima, 1975), the troublesome treatment due to the effect of polarity of radiation is overcome. Moreover, if the surface reflects radiation in accordance with the Lambert law as well as a quite arbitrary phase matrix (Takashima, 1974), the addition method can be easily extended. It is shown in this paper that the addition method is suitable for numerical computation even if the surface reflects light according to the hybrid mode of the diffuse and specular law (Uenoet al., 1974; Mukai, 1976).On leave from the Meterological Research Institute, Tokyo, Japan.     

Scattering and transmission functions of radiation by finite atmospheres with reflecting surfaces

In the present paper, with the aid of invariance principles in connection with the scattering matrix, we get the exact solution of diffuse reflection and transmission problems by finite inhomogeneous, anisotropically scattering atmospheres bounded by reflecting sufaces. On making use of the reflection and transmission integral operators, we show how to obtain the non-linear integro-differential equations for these operators, which do not depend on the initial condition. Then, we have a system of the required integro-differential equations for the scattering and transmission functions. The obtained result is new, so far as we know. Finally, using the scattering matrix, we reduce the diffuse reflection and transmission problems for planetary atmospheres with reflecting surfaces to the standard diffuse reflection and transmission problems.Supported by the National Science Foundation under Grant No. GP29049 and the Atomic Energy Comminission, Division of Research under Contract No. AT(40-3)-133, Project 19.     

Remote method of determining the coordinates of points on a planetary surface

A method and an algorithm for determining the coordinates of points on the planetary surface are described. The coordinates are determined using photographs. To solve the problem, the spacecraft coordinates need to be determined at five trajectory points. The spacecraft trajectory is considered to be a plane. The method is applicable for determining the coordinates of points on the Earth’s surface and on the surface of other planets.     

Periodic orbits of the planetary type and their stability

A review is presented of periodic orbits of the planetary type in the general three-body problem and fourbody problem and the restricted circular and elliptic tnreebody problem. These correspond to planetary systems with one Sun and two or three planets (or a planet and its satellites), the motion of asteoids and also planetary systems with two Suns. The factors which affect the stability of the above configurations are studied in connection with resonance or additional perturbations. Finally, the correspondence of the periodic orbits in the restricted three-body problem with the fixed points obtained by the method of averaging or the method of surface of section is indicated.     

On the Problem of Remote Sensing of the Surface through a Planetary Atmosphere

The problem of remote sensing of the surface through a planetary atmosphere is considered. An efficient approach to the atmospheric correction of satellite information is developed. A model for the atmospheric transfer properties is represented as a linear functional—the superposition integral underlying the classical linear-system approach. The optical transfer operator is constructed mathematically rigorously and physically correctly by the method of influence functions and spatial-frequency characteristics. The influence functions and spatial-frequency characteristics of an atmosphere–planetary surface system are the kernels of the functionals and objective characteristics, which are invariant to specific structures of the objects being sensed and to illumination and observing conditions. The spatial-frequency characteristics are introduced as Fourier transforms of the influence function in horizontal coordinates. The foundations of the spatial-filtering theory are outlined for the problem of remote sensing, which have a wide range of applications. The main problems of the theory and mathematical modeling of three-dimensional radiative transfer are pointed out.     

A new approach of the adding method for the computations of emergent radiation of an inhomogeneous plane-parallel planetary atmosphere

The procedure of computing the intensity and the polarization parameters of radiation diffusely reflected and transmitted by an inhomogeneous, plane-parallel planetary atmosphere is discussed with the aid of the adding method. If the atmosphere is simulated by a number of homogeneous sublayers (aerosols and ozone may be included), the matrices of radiation diffusely reflected and transmitted by the atmosphere can be expressed in terms of these matrices of sublayers by using only a couple of iterative equations with the polarity effect of radiation. This procedure is to be extended to the model atmosphere bounded by the surface reflector with a quite arbitrary phase matrix.     

Albedo-shift method in the problem of anisotropic light scattering in a plane atmosphere

A standard problem of radiative transfer theory — calculating the diffuse reflection and transmission of radiation by a plane scattering atmosphere — is considered. The recently proposed albedoshift method is used to calculate the X and Y functions (and the H function) for the case of anisotropic scattering with a Henyey-Greenstein indicatrix. The method enables one to “suppress” scattering and obtain iterative solutions of high accuracy in only a few iterations, even when the mean number of photon scatterings in the atmosphere is very large. Translated from Astrofizika, Vol. 41. No. 4, pp. 623–646, October–December, 1998.     

Characteristics of the radiation emerging from the top of a Rayleigh atmosphere—II Total upward flux and albedo

The solution of the equation of radiative transfer in a medium exhibiting Rayleigh scattering, as developed by S. Chandrasekhar, has been used for an extensive series of computations(³) of the characteristics of the scattered and diffusely reflected radiation emerging from the top of an atmospheric model which corresponds in many respects to the sunlit portion of the earth's atmosphere. The first part of this two-part discussion dealt with the intensity, degree of polarization, plane of polarization and the neutral points of the emergent light as functions of sun elevation, direction in the downward hemisphere, optical thickness of the model atmosphere and reflectivity of the underlying surface. This second part is concerned with the upward flux obtained by an integration of the intensity over the entire hemisphere, for the incident radiation (a) being independent of wavelength or (b) having the spectral distribution of the extra-terrestrial solar radiation. Integration with respect to wavelength in the latter case, together with an approximation for the sphericity of the atmosphere, yields a value of 7.6 per cent for the earth's planetary albedo due to scattering by the clear atmosphere. An approximation for ozone absorption decreases the computed albedo to 6.9 per cent.     

Derivation of planetary topography using multi-image shape-from-shading

In many cases, the derivation of high-resolution digital terrain models (DTMs) from planetary surfaces using conventional digital image matching is a problem. The matching methods need at least one stereo pair of images with sufficient texture. However, many space missions provide only a few stereo images and planetary surfaces often possess insufficient texture.This paper describes a method for the generation of high-resolution DTMs from planetary surfaces, which has the potential to overcome the described problem. The suggested method, developed by our group, is based on shape-from-shading using an arbitrary number of digital optical images, and is termed “multi-image shape-from-shading” (MI-SFS). The paper contains an explanation of the theory of MI-SFS, followed by a presentation of current results, which were obtained using images from NASA's lunar mission Clementine, and constitute the first practical application with our method using extraterrestrial imagery. The lunar surface is reconstructed under the assumption of different kinds of reflectance models (e.g. Lommel-Seeliger and Lambert). The represented results show that the derivation of a high-resolution DTM of real digital planetary images by means of MI-SFS is feasible.     

Direct perturbations of the planets on the Moon's motion

The aim of the present paper is to present the theoretical background of a method to compute the planetary perturbations on the Moon's motion. We formulate an algorithm based upon the Lie transform method and well-suited to the particular problem at hand.This algorithm is being implemented using Henrard's Semi-Analytical Lunar Ephemeris (SALE) as solution of the Main Problem and Bretagnon's planetary theory. The accuracy of the solution is intended to be about 0".001 for terms of period up to 2000 years.To illustrate the interest of our approach, we comment on some preliminary results obtained about the direct perturbations due to Venus on the Moon's longitude. The final results will be the subject of another paper.     

Planetary perturbations on the Libration of the Moon

A theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field, was recently built (Moons, 1982). The Lie transforms method was used to reduce the Hamiltonian of the main problem of the libration of the Moon and to produce the usual libration series p₁, p₂ and . This main problem takes into account the perturbations due to the Sun and the Earth on the rotation of a rigid Moon about its center of mass. In complement to this theory, we have now computed the planetary effects on the libration, the planetary terms being added to the mean Hamiltonian of the main problem before a last elimination of the angles. For the main problem, as well as for the planetary perturbations, the motion of the center of mass of the Moon is described by the ELP 2000 solution (Chapront and Chapront-Touze, 1983).     

Radiative Transfer in Inhomogeneous Atmospheres. I

This series of papers is devoted to multiple scattering of light in plane parallel, inhomogeneous atmospheres. The approach proposed here is based on Ambartsumyan's method of adding layers. The main purpose is to show that one can avoid difficulties with solving various boundary value problems in the theory of radiative transfer, including some standard problems, by reducing them to initial value problems. In this paper the simplest one dimensional problem of diffuse reflection and transmission of radiation in inhomogeneous atmospheres with finite optical thicknesses is considered as an example. This approach essentially involves first determining the reflection and transmission coefficients of the atmosphere, which, as is known, are a solution of the Cauchy problem for a system of nonlinear differential equations. In particular, it is shown that this system can be replaced with a system of linear equations by introducing auxiliary functions P and S. After the reflectivity and transmissivity of the atmosphere are determined, the radiation field in it is found directly without solving any new equations. We note that this approach can be used to obtain the required intensities simultaneously for a family of atmospheres with different optical thicknesses. Two special cases of the functional dependence of the scattering coefficient on the optical thickness, for which the solutions of the corresponding equations can be expressed in terms of elementary functions, are examined in detail. Some numerical calculations are presented and interpreted physically to illustrate specific features of radiative transport in inhomogeneous atmospheres.     

Exact solution of a key equation in a finite planetary atmosphere by the method of laplace transform and linear singular operators

Considering the ground reflection according to Lambert's law, we establish a fundamental equation in finite planetary atmospheres. An exact form of the solution of this equation is obtained for the emergent quantities from the bounding faces in terms ofX-Y equations by the method of Laplace transform, in combination with the theory of linear singular operators.     

Negative ions in the ionospheres of planetary bodies without atmospheres

Negative ions may be formed in the ionospheres of Mercury, the Moon and Jupiter's satellites with densities about a few % of the ionospheric electron density. The negative ions are produced by three mechanisms at the planetary surface: charge inversion during energetic proton scattering, with simultaneous secondary negative ion emission, and micrometeorite impacts. The density and distribution of negative ions around planetary bodies depends primarily upon the negative ion life-times determined by photodetachment by solar radiation.     

Scattering of Light by Composite Particles in a Planetary Surface

It has been proposed that composite particles containing internal scatterers may provide the explanation for the fact that most photometric studies of planetary surfaces based on Hapke's model of bidirectional reflectance have found the planetary particles to exhibit moderately backscattering phase functions. However, an implicit assumption made in this explanation is that the scattering by composite particles containing multiple internal inclusions in a planetary surface can still be adequately computed using standard radiative transfer theory assuming the composite particles to be the fundamental individual scatterers even though such particles are necessarily in close proximity to each other. In this paper, this assumption is explored by examining the effects of close packing on the light scattering by spherical particles containing isotropic internal scatterers using a Monte Carlo routine. As expected, classical radiative transfer (assuming a random distribution of scattering particles) coupled with the assumption that the composite particle is the fundamental scatterer provides a good approximation in the high porosity limit. However, even for porosities as high as 90% the effects of close packing are clearly seen with the radiative transfer calculation underestimating the scattering by ∼10% at high incidence, emission, and phase angles. As the porosity is lowered further, the discrepancy becomes more severe and can reach 50% or more. In contrast, assuming the individual scatterer properties in the radiative transfer calculation leads to a substantial overestimate of the scattering even for porosities as low as 27.5%. This suggests that parameters derived using the classical radiative transfer theory will yield results intermediate between those of the composite as a whole and those of the internal scatterers. Thus, one should exercise caution in interpreting the results of models based on classical radiative transfer theory in terms of the physical properties of the surface particles and, where possible, the bidirectional reflectance of densely packed composite particles should be computed using more accurate methods such as the stochastic radiative transfer theory.